| 1. |
- Barrio, Roberto, et al.
(författare)
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A database of rigorous and high-precision periodic orbits of the Lorenz model
- 2015
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Ingår i: Computer Physics Communications. - : Elsevier BV. - 0010-4655 .- 1879-2944. ; 194, s. 76-83
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Tidskriftsartikel (refereegranskat)abstract
- A benchmark database of very high-precision numerical and validated initial conditions of periodic orbits for the Lorenz model is presented. This database is a "computational challenge" and it provides the initial conditions of all periodic orbits of the Lorenz model up to multiplicity 10 and guarantees their existence via computer-assisted proofs methods, The orbits are computed using high-precision arithmetic and mixing several techniques resulting in 1000 digits of precision on the initial conditions of the periodic orbits, and intervals of size 10100 that prove the existence of each orbit. Program summary Program title: Lorenz-Database Catalogue identifier: AEWM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEWM_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8515 No. of bytes in distributed program, including test data, etc.: 6964501 Distribution format: tar.gz Programming language: Data. Computer: Any computer. Operating system: Any. RAM: Database, no requirements Classification: 4.3, 4.12. Nature of problem: Database of all periodic orbits of the Lorenz model up to multiplicity 10 with 1000 precision digits. Solution method: Advanced search methods for locating unstable periodic orbits combined with the Taylor series method for multiple precision integration of ODEs and interval methods for providing Computer-Assisted proofs of the periodic orbits. Unusual features: The database gives 100 digits rigorously proved using Computer-Assisted techniques and 1000 digits using an optimal adaptive Taylor series method. Running time: Not Applicable.
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| 2. |
- Bartha, Ferenc A., et al.
(författare)
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Fixed points of a destabilized Kuramoto-Sivashinsky equation
- 2015
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Ingår i: Applied Mathematics and Computation. - : Elsevier BV. - 0096-3003 .- 1873-5649. ; 266, s. 339-349
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Tidskriftsartikel (refereegranskat)abstract
- We consider the family of destabilized Kuramoto-Sivashinsky equations in one spatial dimension u(t) + nu u(xxx) + beta u(xx) + gamma uu(x) - alpha u for alpha, nu >= 0 and beta, gamma is an element of R. For certain parameter values, shock like stationary solutions have been numerically observed. In this work we verify the existence of several such solutions using the framework of self consistent bounds and validated numerics. Published by Elsevier Inc.
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| 3. |
- Belova, Anna, 1987-
(författare)
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Computational dynamics – real and complex
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts.In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number.In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes).Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients.Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.
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| 4. |
- Dahne, Joel, et al.
(författare)
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Enclosing all zeros of a system of analytic functions
- 2019
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Ingår i: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 348, s. 513-522
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Tidskriftsartikel (refereegranskat)abstract
- We present a rigorous numerical method for location of simple zeros of a system of two analytic functions in a rectangular cuboid domain based on the logarithmic integral. We compare this to a simpler, also rigorous, method based on bisection. The latter is determined to be more efficient in the examples considered. This is mainly due to inefficient methods for computing the logarithmic integral occurring in the former method. (C) 2018 Elsevier Inc. All rights reserved.
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| 5. |
- Galias, Zbigniew, et al.
(författare)
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Is the Hénon attractor chaotic?
- 2015
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Ingår i: Chaos. - : AIP Publishing. - 1054-1500 .- 1089-7682. ; 25:3
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Tidskriftsartikel (refereegranskat)abstract
- By performing a systematic study of the Henon map, we find low-period sinks for parameter values extremely close to the classical ones. This raises the question whether or not the well-known Henon attractor-the attractor of the Henon map existing for the classical parameter values-is a strange attractor, or simply a stable periodic orbit. Using results from our study, we conclude that even if the latter were true, it would be practically impossible to establish this by computing trajectories of the map.
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| 6. |
- Galias, Zbigniew, et al.
(författare)
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On the Existence of the Double Scroll Attractor for the Chua's Circuit with a Smooth Nonlinearity
- 2018
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Ingår i: 2018 IEEE INTERNATIONAL SYMPOSIUM ON CIRCUITS AND SYSTEMS (ISCAS). - : IEEE. - 9781538648810
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Konferensbidrag (refereegranskat)abstract
- In simulations of the Chua's circuit with a smooth nonlinearity for certain parameter values one observes the double scroll attractor. This attractor contains an unstable equilibrium, and typical trajectories belonging to the attractor may pass arbitrarily close to this equilibrium. In consequence, it is impossible to compute trajectories over the whole attractor using standard rigorous numerical integration procedures. This is due to the existence of trajectories which spend arbitrarily long time in a neighborhood of the equilibrium. In this work, a method to find enclosures of trajectories passing arbitrarily close to an unstable fixed point of spiral type is presented. This method is used to prove the existence of a trapping region enclosing the double scroll attractor for the Chua's circuit with a cubic nonlinearity.
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| 7. |
- Galias, Zbigniew, et al.
(författare)
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Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor
- 2019
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 266:5, s. 2408-2434
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.
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| 8. |
- Joldes, Mioara, et al.
(författare)
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CAMPARY : Cuda Multiple Precision Arithmetic Library and Applications
- 2016
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Ingår i: Mathematical Software, ICMS 2016. - Cham : Springer International Publishing. - 9783319424323 - 9783319424316 ; , s. 232-240
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Konferensbidrag (refereegranskat)abstract
- Many scientific computing applications demand massive numerical computations on parallel architectures such as Graphics Processing Units (GPUs). Usually, either floating-point single or double precision arithmetic is used. Higher precision is generally not available in hardware, and software extended precision libraries are much slower and rarely supported on GPUs. We develop CAMPARY: a multipleprecision arithmetic library, using the CUDA programming language for the NVidia GPU platform. In our approach, the precision is extended by representing real numbers as the unevaluated sum of several standard machine precision floating-point numbers. We make use of error-free transforms algorithms, which are based only on native precision operations, but keep track of all rounding errors that occur when performing a sequence of additions and multiplications. This offers the simplicity of using hardware highly optimized floating-point operations, while also allowing for rigorously proven rounding error bounds. This also allows for easy implementation of an interval arithmetic. Currently, all basic multiple-precision arithmetic operations are supported. Our target applications are in chaotic dynamical systems or automatic control.
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| 9. |
- Mitrea, Irina, et al.
(författare)
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Invertibility Properties of Singular Integral Operators Associated with the Lam, and Stokes Systems on Infinite Sectors in Two Dimensions
- 2017
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Ingår i: Integral equations and operator theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 89:2, s. 151-207
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on , , whenever is an infinite sector in . This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lam, system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector and the differential operator in question.
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| 10. |
- Ryeznik, Yevgen, 1979-
(författare)
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Optimal adaptive designs and adaptive randomization techniques for clinical trials
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this Ph.D. thesis, we investigate how to optimize the design of clinical trials by constructing optimal adaptive designs, and how to implement the design by adaptive randomization. The results of the thesis are summarized by four research papers preceded by three chapters: an introduction, a short summary of the results obtained, and possible topics for future work.In Paper I, we investigate the structure of a D-optimal design for dose-finding studies with censored time-to-event outcomes. We show that the D-optimal design can be much more efficient than uniform allocation design for the parameter estimation. The D-optimal design obtained depends on true parameters of the dose-response model, so it is a locally D-optimal design. We construct two-stage and multi-stage adaptive designs as approximations of the D-optimal design when prior information about model parameters is not available. Adaptive designs provide very good approximations to the locally D-optimal design, and can potentially reduce total sample size in a study with a pre-specified stopping criterion.In Paper II, we investigate statistical properties of several restricted randomization procedures which target unequal allocation proportions in a multi-arm trial. We compare procedures in terms of their operational characteristics such as balance, randomness, type I error/power, and allocation ratio preserving (ARP) property. We conclude that there is no single “best” randomization procedure for all the target allocation proportions, but the choice of randomization can be done through computer-intensive simulations for a particular target allocation.In Paper III, we combine the results from the papers I and II to implement optimal designs in practice when the sample size is small. The simulation study done in the paper shows that the choice of randomization procedure has an impact on the quality of dose-response estimation. An adaptive design with a small cohort size should be implemented with a procedure that ensures a “well-balanced” allocation according to the D-optimal design at each stage.In Paper IV, we obtain an optimal design for a comparative study with unequal treatment costs and investigate its properties. We demonstrate that unequal allocation may decrease the total study cost while having the same power as traditional equal allocation. However, a larger sample size may be required. We suggest a strategy on how to choose a suitable randomization procedure which provides a good trade-off between balance and randomness to implement optimal allocation. If there is a strong linear trend in observations, then the ARP property is important to maintain the type I error and power at a certain level. Otherwise, a randomization-based inference can be a good alternative for non-ARP procedures.
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